Determine if Σ lnx/√x dx converges or diverges

->Show the series satisfies the hypothesis of the integral test.
->Use the Integral Test to determine if the series converges or diverges.

Assuming that x ≥ 1, the integrand is nonnegative (and continuous) for all x ≥ 1.

Now, we evaluate ∫(1 to ∞) ln x dx/x^(1/2).

Using integration by parts with
u = ln x, dv = x^(-1/2) dx
dv = dx/x, v = 2x^(1/2)

So, we get
2x^(1/2) ln x {for x = 1 to ∞} - ∫(1 to ∞) 2 dx/x^(1/2)
= [2x^(1/2) ln x - 4x^(1/2)] {for x = 1 to ∞}
= (2 ln x - 4) * x^(1/2) {for x = 1 to ∞}
= ∞.

Hence, the integral is divergent.

I hope this helps!